TY - JOUR
T1 - Bi-invariant metrics and quasi-morphisms on groups of Hamiltonian diffeomorphisms of surfaces
AU - Brandenbursky, Michæl
N1 - Funding Information:
The author was partially supported by the CRM-ISM fellowship. He would like to thank CRM-ISM Montreal for the support and great research atmosphere.
Funding Information:
The author would like to thank the anonymous referee for useful comments and remarks. He also would like to thank Mladen Bestvina, Danny Calegari, Louis Funar, Juan Gonzalez-Meneses, Jarek Kedra, Chris Leininger, Dan Margalit and Pierre Py for fruitful discussions. Part of this work has been done during the author’s stay in Mathematisches Forschungsinstitut Oberwolfach and Max Planck Institute for Mathematics in Bonn. The author wishes to express his gratitude to both institutes. He was supported by the Oberwolfach Leibniz fellowship and Max Planck Institute research grant.
Publisher Copyright:
© 2015 World Scientific Publishing Company.
PY - 2015/8/29
Y1 - 2015/8/29
N2 - Let σg be a closed orientable surface of genus g and let Diff0(σg, area) be the identity component of the group of area-preserving diffeomorphisms of σg. In this paper, we present the extension of Gambaudo-Ghys construction to the case of a closed hyperbolic surface σg, i.e. we show that every nontrivial homogeneous quasi-morphism on the braid group on n strings of σg defines a nontrivial homogeneous quasi-morphism on the group Diff0 (σg, area). As a consequence we give another proof of the fact that the space of homogeneous quasi-morphisms on Diff0(σg, area) is infinite-dimensional. Let Ham(σg) be the group of Hamiltonian diffeomorphisms of σg. As an application of the above construction we construct two injective homomorphisms Zm → Ham(σg), which are bi-Lipschitz with respect to the word metric on Zm and the autonomous and fragmentation metrics on Ham(σg). In addition, we construct a new infinite family of Calabi quasi-morphisms on Ham(σg).
AB - Let σg be a closed orientable surface of genus g and let Diff0(σg, area) be the identity component of the group of area-preserving diffeomorphisms of σg. In this paper, we present the extension of Gambaudo-Ghys construction to the case of a closed hyperbolic surface σg, i.e. we show that every nontrivial homogeneous quasi-morphism on the braid group on n strings of σg defines a nontrivial homogeneous quasi-morphism on the group Diff0 (σg, area). As a consequence we give another proof of the fact that the space of homogeneous quasi-morphisms on Diff0(σg, area) is infinite-dimensional. Let Ham(σg) be the group of Hamiltonian diffeomorphisms of σg. As an application of the above construction we construct two injective homomorphisms Zm → Ham(σg), which are bi-Lipschitz with respect to the word metric on Zm and the autonomous and fragmentation metrics on Ham(σg). In addition, we construct a new infinite family of Calabi quasi-morphisms on Ham(σg).
KW - Groups of Hamiltonian diffeomorphisms
KW - bi-invariant metrics
KW - braid groups
KW - mapping class groups
KW - quasi-morphisms
UR - http://www.scopus.com/inward/record.url?scp=84940459276&partnerID=8YFLogxK
U2 - 10.1142/S0129167X15500664
DO - 10.1142/S0129167X15500664
M3 - Article
AN - SCOPUS:84940459276
SN - 0129-167X
VL - 26
JO - International Journal of Mathematics
JF - International Journal of Mathematics
IS - 9
M1 - 1550066
ER -